Let B = V₁,..., V₁ be a maximal linearly independent list in a vector space V in the sense that adding another element of V to the list would make it linearly dependent. Does it follow that B is a basis? Justify your answer. Select one: O No, since we did not specify that Vis finite-dimensional. It could be infinite-dimensional and hence not admit a basis of size n = = |B| for any n O No, since we did not specify the field. If n is a prime and we work over Fr, then we can get a counterexample O No, since the list is maximal, it must be that n = ∞ because for any finite n we could always add Un+1 chosen to be linearly independent of the v₁, ..., Vn. But then if V was finite-dimensional we would have a contradiction. So its not always possible. O Yes, for any v € V, the list v, v₁, ···, Vn is l.d. so cv + €₁V₁ + ... + Cnºn = 0 for some c, c; in the field not all zero. But c = 0 else the V₁,...,Un would have a linear relation, hence we can divide by c and write v = ₁(-)v₁, so v₁, ..., vn also span, hence a basis. O None of the others apply

Let B = V₁,..., V₁ be a maximal linearly independent list in a vector space V in the sense that adding another element of V to the list would make it linearly dependent. Does it follow that B is a basis? Justify your answer. Select one: O No, since we did not specify that Vis finite-dimensional. It could be infinite-dimensional and hence not admit a basis of size n = = |B| for any n O No, since we did not specify the field. If n is a prime and we work over Fr, then we can get a counterexample O No, since the list is maximal, it must be that n = ∞ because for any finite n we could always add Un+1 chosen to be linearly independent of the v₁, ..., Vn. But then if V was finite-dimensional we would have a contradiction. So its not always possible. O Yes, for any v € V, the list v, v₁, ···, Vn is l.d. so cv + €₁V₁ + ... + Cnºn = 0 for some c, c; in the field not all zero. But c = 0 else the V₁,...,Un would have a linear relation, hence we can divide by c and write v = ₁(-)v₁, so v₁, ..., vn also span, hence a basis. O None of the others apply

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Suppose we have a set of three vectors S = {V₁, V2, V3} C R². Which one of the following is a true statement? (a) S will be a basis for R2 only if one of the vectors is the zero vector. (b) S will be a basis for R2 if and only if v₁, V2, and v3 are linearly independent. (c) S will be a basis for R2 as long as it is possible to write V3 = C1V1 + C2V2. (d) S cannot be a basis for R2 because 3 vectors cannot span R². (e) S cannot be a basis for R2 because 3 vectors cannot be linearly independent in R².
Given a basis B {V1, V2, · .., vn} of Subspace S, there is a unique way to express any v E S as a s linear combinations of {v1, V2, · · · , Vn}.
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se E Exercise 1.1.3 Show that in any linear space V, a set of vectors is always linearly dependent if one of the vectors is zero.
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